Any message you send on a channel could be logged by any user or bot in that channel. That depends on both the policy of the channel, and on the settings and capabilities of the IRC clients of the individual users of that channel.
If you type something in a public channel, there's no guarantee it will remain private, you have no control over what other people do with those messages. You may want to use search engines to search for channel logs to make sure. And check the channel's website if there is any.
There's no reason why private messages couldn't be logged, it depends on the preference of the user and on whether their client supports it. Of course, they could always manually cut and paste too, if their client doesn't support logging automatically. I don't recall ever having seen anyone put private logs online, at least not without permission.
Normal private messages go from the sender's client to the sender's server, via the network to the receiver's server, to the receiver's client. So in theory it could be logged by the servers connected to the network, but I don't see that ending up in a public log.
With DCC CHAT, the IRC network is used to set up the connection, but once it's established, you're no longer going through the network. The messages are not encrypted, so in theory they could be intercepted at any point between you and the receiver. However, the odds of that happening are probably very small.